BIDDING
BIDDING CHESS
CHESS
JAY
PAYNE
JAY BHAT
BHAT AND
AND SAM PAYNE
allstarted
startedwith
withaachessboard
chessboard
ItItall
8
7
6
5
4
3
2
1
a
b
c
d
e
f
g
h
andaabottle
bottleofofraki
raki
and
at an otherwise respectable educational institution. SP’s friend Ed had just returned
atfrom
an otherwise
respectable
institution.
SP’sand
friend
Edsheets
had just
returned
Turkey with
a bottle ofeducational
the good stuff.
We were two
a half
to the
wind,
from
Turkey
with
a
bottle
of
the
good
stuff.
We
were
two
and
a
half
sheets
to
the
wind,
feeling oppressed by the freshman dormroom, its cinderblock walls, and our mediocre
feeling
oppressed
the freshman
dormroom,
cinderblock
walls, and
and time,
our mediocre
chess skills.
The by
details
of what transpired
areits
clouded
with alcohol
but Ed
chess
skills.
The
details
of
what
transpired
are
clouded
with
alcohol
and
time,emerged
but Ed
was enrolled in a seminar on auction theory and a site called eBay had recently
was
a seminar
auctionthat
theory
and a site
called
eBay
had recently
emerged
on enrolled
the web. inSomehow
weondecided
alternating
moves
was
boring—too
predictable.
onSome
the web.
Somehow
we
decided
that
alternating
moves
was
boring—too
predictable.
moves are clearly worth more than others, and the game would be much more
Some
movesifare
than to
others,
theBidding
game would
interesting
youclearly
had toworth
pay formore
the right
move.and
Enter
Chess. be much more
interesting
if
you
had
to
pay
for
the
right
to
move.
Enter
Bidding
Chess.
How to play. We both start with one hundred chips. Before each
move, we write
How
to
play.
We
both
start
with
one
hundred
chips.
Before
each
move,
write
down our bids, and the player who bids more gives that many chips to the
otherweplayer
down our bids, and the player who bids more
1 gives that many chips to the other player
1
2
BHAT AND PAYNE
BHAT AND PAYNE
2
and makes a move on the chessboard. For example, if I bid nineteen for the first move
andyou
makes
move on thethen
chessboard.
if I chips
bid nineteen
for athe
first on
move
and
bid atwenty-four,
you giveFor
me example,
twenty-four
and make
move
the
and you bidNow
twenty-four,
then
youand
give
mehave
twenty-four
chips
make
a move
on the
chessboard.
I have 124
chips
you
76 and we
bidand
for the
second
move.
By
chessboard.
Now
I
have
124
chips
and
you
have
76
and
we
bid
for
the
second
move.
By
the way, you bid way too much and now you’re toast!
the
way,toyou
bid way
and now you’re
toast!
How
win.
You too
winmuch
by capturing
your opponent’s
king. Or, rather, I win by
How to win. You win by capturing your opponent’s king. Or, rather, I win by
capturing your king. None of this woo-woo checkmate stuff. I don’t care if you have me
capturing your king. None of this woo-woo checkmate stuff. I don’t care if you have me
in checkmate when I have enough chips to make seven moves in a row.
in checkmate when I have enough chips to make seven moves in a row.
What if the bids are tied? Quibblers. The bids are never tied. There is an extra
What if the bids are tied? Quibblers. The bids are never tied. There is an extra
chip, called *. If you have * in your pile, then you can include it with your bid and win
chip, called *. If you have * in your pile, then you can include it with your bid and win
any tie. And if you win, then * goes to your opponent along with the rest of your bid.
any tie. And if you win, then * goes to your opponent along with the rest of your bid.
But
tie.
Butififyou
youwuss
wussout
outand
andsave
save** for
for later,
later, then
then you
you lose
lose the
the tie.
Bidding
Chess
is
meant
to
be
played,
so
set
up
the
board
and grab
grab aa friend,
friend, maybe
maybe
Bidding Chess is meant to be played, so set up the board and
one
Think carefully—-the
carefully—-the game
game
oneyou
younever
neverreally
reallyliked
liked much
much anyway,
anyway, and
and try
try it
it out.
out. Think
isisnever
After you
you have
have played
played aa few
few
neverwon
wonon
onthe
thefirst
firstmove,
move, but
but itit isis often
often lost
lost there.
there. After
times,
take
a
look
at
the
following
transcript
from
a
game
played
in
1015
Evans
Hall,
times, take a look at the following transcript from a game played in 1015 Evans Hall,
the
in fall
fall 2006.
2006. Names
Names have
havebeen
been
thecommon
commonroom
roomofofthe
theUC
UCBerkeley
Berkeley math
math department,
department, in
changed
for
reasons
you
may
imagine.
We
write
N
*
to
denote
a
pile
or
bid
with
N
chips
changed for reasons you may imagine. We write N * to denote a pile or bid with N chips
plus
plusthe
the* *chip.
chip.
AAsample
chips. Alice
Alice offers
offersBob
Bob
samplegame.
game.Alice
Aliceand
andBob
Bob both
both start
start with
with one
one hundred
hundred chips.
the
*
chip,
but
he
refuses.
Alice
shrugs,
takes
the
black
pieces,
and
bids
twelve
for
the
the * chip, but he refuses. Alice shrugs, takes the black pieces, and bids twelve for the
first
firstmove.
move.Bob
Bobbids
bidsthirteen
thirteenand
and moves
moves his
his knight
knight to c3.
8
7
6
5
4
3
2
1
a
b
c
d
e
f
g
h
Now
NowAlice
Alicehas
has113*
113*chips,
chips,and
andBob
Bobhas
has87
87as
asthey
theyponder
ponderthe
thevalue
valueofofthe
thenext
nextmove.
move.
Alice
figures
that
the
second
move
cannot
possibly
be
worth
more
than
the
first,
because
Alice figures that the second move cannot possibly be worth more than the first, because
wouldbebesilly
sillytotobid
bid more
more than
than thirteen
thirteen and
and end
end up
up in
ititwould
in aa symmetric
symmetric position
position with
with
fewer
chips
than
Bob.
So
she
bids
eleven
plus
*,
which
feels
about
right.
fewer chips than Bob. So she bids eleven plus *, which feels about right. Bob
Bob reasons
reasons
similarly,and
andalso
alsobids
bids eleven.
eleven. Alice
Alice wins
wins the
the tie
tie with
with ** and
similarly,
and moves
moves her
her king’s
king’s pawn
pawn
forwardone
onetotoe6.
e6.
forward
BIDDING CHESS
CHESS
BIDDING
33
BIDDING CHESS
3
Bob, who
who played
played competitive
competitive chess
chess in
in high
high school,
school, is
is puzzled
puzzled by
by this
this conservative
Bob,
conservative
opening
move.
Feeling
comfortable
with
his
board
position,
he
decides
to
bid
only
Bob, move.
who played
school,
is puzzled
by this
conservative
opening
Feelingcompetitive
comfortablechess
withinhishigh
board
position,
he decides
to bid
only nine
nine
of
his
98*
chips
for
the
third
move
and
is
mildly
surprised
when
Alice
bids
fifteen.
She
comfortable
withishis
boardsurprised
position,when
he decides
bidfifteen.
only nine
ofopening
his 98*move.
chips Feeling
for the third
move and
mildly
Alice to
bids
She
moves
her
bishop
to
c5.
of his her
98*bishop
chips for
moves
to the
c5. third move and is mildly surprised when Alice bids fifteen. She
moves her bishop to c5.
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
a
a
b
b
c
c
d
d
e
e
f
f
g
g
h
h
NowAlice
Alicehas
has8787chips,
chips,
while
Bob
113*.
Since
Alice
15 the
for last
the move
last move
Now
while
Bob
hashas
113*.
Since
Alice
bid bid
15 for
and
Now Alice has 87 chips, while Bob has 113*. Since Alice bid 15 for the last move
and
started
attack
that
would
like
counter,Bob
Bobbids
bidsfifteen
fifteenfor
for the
the next
next move,
started
an an
attack
that
he he
would
like
totocounter,
and started an attack that he would like to counter, Bob bids fifteen for the next move,
whichseems
seemsfair.
fair. Alice
Alicebids
bids twenty-two
twenty-two and
and takes
takes the
the pawn
pawn at f2.
which
which seems fair. Alice bids twenty-two and takes the pawn at f2.
Bobrealizes
realizeswith
withsome
somedismay
dismay that
that he
he must
must win
win the
the next
next move to prevent Alice from
Bob
Bob realizes with some dismay that he must win the next move to prevent Alice from
taking
his
kind.
She
bids
all
65
of
her
chips,
and
he
bids
65*. King
Kingtakes
takesbishop.
bishop.
taking
he bids
bids 65*.
takinghis
hiskind.
kind. She
Shebids
bids all
all 65
65 of
of her
her chips,
chips, and
and he
88
77
66
55
44
33
22
11
aa
bb
cc
d
e
f
gg
hh
Now
Bob
has
material
advantage,
chips.
Pondering
Now
Bob
has
aamaterial
advantage,
Alice
130*
chips.
Pondering
theboard,
board,
Now
Bob
has
a material
advantage,
butbut
Alice
hashas
130*130*
chips.
Pondering
the the
board,
Bob
Bob
realizes
that
if
Alice
wins
the
bid
for
less
than
thirty,
then
she
can
move
her
queen
Bob
realizes
that
if
Alice
wins
the
bid
thirty,
then
she
can
move
her
queen
realizes that if Alice wins the bid for less than thirty, then she can move her queen out
outf6
tothreaten
threaten
his
king
and
then
win
the
next
out
tototo
f6f6to
king
and
then
bideverything
everythingtoto
towin
winthe
thenext
nextmove
moveand
andtake
takehis
his
to
threaten
hishis
king
and
then
bid
move
and
take
his
king.
So
Bob
bids
thirty,
winning
over
Alice’s
bid
of
twenty-five.
He
moves
his
knight
king.
his knight
knight
king. So
SoBob
Bobbids
bids thirty,
thirty, winning
winning over
over Alice’s bid of twenty-five.
twenty-five. He
He moves
moves his
tof3.
f3.
But
Alice
can
still
threaten
Bob’s
by
moving
her
queen
to
h5.
Since
toto
But
Alice
can
still
threaten
Bob’s
king
moving
her
queen
to
h5.
Since
she
f3. But Alice can still threaten Bob’s king by moving her queen to h5. Sinceshe
she
has
160*
chips,
she
can
now
win
the
next
two
moves,
regardless
of
what
Bob
bids,
and
has 160* chips, she can now win the next
regardless of what Bob bids, and
4
BHAT AND PAYNE
has 160* chips, she can now win the next two moves, regardless of what Bob bids, and
capture his king. Alice suppresses a smile as Bob realizes he has been defeated. Head
in his hands, he mumbles, “That was a total mindf**k.”
Richman’s Theory. David Richman invented and studied a class of similar bidding
games in the late 1980s in which bids are allowed to be arbitrary nonnegative real
numbers, not just integers. One of Richman’s discoveries is a surprising connection
between such bidding games and random-turn games in which, instead of alternating
moves, players flip a fair coin to determine who moves next.1 For simplicity, suppose
Alice and Bob are playing a finite, loop-free combinatorial game G. Let P (G) be the
probability that Alice wins, assuming optimal play, and let R(G) be the critical threshold
between zero and one such that Alice wins if her proportion of the bidding chips is more
than R(G) and Bob wins if she has less than R(G), with real-valued bidding.
Richman’s Theorem. Let G be a finite combinatorial game. Then
R(G) = 1 − P (G).
Furthermore, a move is optimal for random-turn play if and only if it is an optimal
move with real-valued bidding. The proof of Richman’s Theorem is disarmingly simple;
one shows that R(G) and 1 − P (G) satisfy the same recursion with the same initial
conditions, as follows. For any position v in the game G, let Gv be the game played
starting from v.
Proof. Suppose v is an ending position, so it is a winning position for either Alice or
Bob. If v is a winning position for Alice, then R(G) and 1 − P (G) are both equal to
zero; if v is a winning position for Bob, then both are equal to one. Suppose v is not an
ending position. By induction on the length of the game, we may assume that R(Gw )
is equal to 1 − P (Gw ) for every position w that can be moved to from v. Let R+ (v)
be the maximum value of R(Gw ), over all positions w that Bob can move to from v,
and let R− (v) be the minimum over all positions that Alice can move to. Then it is
straightforward to check that
R+ (v) + R− (v)
R(Gv ) =
,
2
and an optimal bid for both players is R+ (v) − R− (v) 2, since Alice will always move
to a position that minimizes R, and Bob will move to a position that maximizes it.
Similarly, Alice will always move from v to a position that minimizes 1 − P , and Bob
will move to a position that maximizes it. These probabilities are equal to R+ (v) and
R− (v), by induction, and 1 − P (Gv ) is the average of the two, since we flip a fair coin
to determine who moves next.
1The
details of this result, and many other related facts, were presented by Richman’s friends and
collaborators in [?, ?]
BIDDING CHESS
5
Real vs. discrete bidding. Real valued bids are convenient for theoretical purposes, and are essential to Richman’s elegant theory. They are, however, disastrous
for
√
recreational play, as becomes obvious when one player bids something like e− π + log 17.
Whole number bids are playable and fun, but the general theory and practical computation of optimal strategies become much more subtle. For instance, the set of optimal
first moves for Tic-Tac-Toe with discrete bidding depends on the number of chips in
play [?]. Nevertheless, when the number of chips is large, discrete bidding approximates
continuous bidding well enough for many purposes, and Richman’s theory gives deep
insight into discrete bidding game play.
Bidding Hex. Richman’s Theorem is especially exciting in light of recent developments in the theory of random-turn games. The probabilists Peres, Schramm, Sheffield,
and Wilson found an elegant solution to Random-Turn Hex, along with a Monte Carlo
algorithm that quickly produces optimal or near-optimal moves [?]. This algorithm has
been implemented by David Wilson [?], and the computer beats a skilled human opponent more than half the time, though anyone can beat it sometimes, by winning enough
coin flips. Elina Robeva has implemented a similar algorithm for Bidding Hex that is
overwhelmingly effective—undefeated against human opponents.
Online. See the Secret Blogging Seminar post by Noah Snyder [?] for an excellent
online introduction to bidding games, and links to further resources. JB has developed
Bidding Tic-Tac-Toe and Bidding Hex for online play through Facebook. You can challenge your friends to a game of skill and honor, or play against the computer at a range
of difficulty settings. Visit
http://apps.facebook.com/biddingttt and http://apps.facebook.com/biddinghex
and try it out!
References
[DP08]
M. Develin and S. Payne, Discrete bidding games, preprint, 2008.
[LLP+ 99] A. Lazarus, D. Loeb, J. Propp, W. Stromquist, and D. Ullman, Combinatorial games under
auction play, Games Econom. Behav. 27 (1999), no. 2, 229–264.
[LLPU96] A. Lazarus, D. Loeb, J. Propp, and D. Ullman, Richman games, Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol. 29, Cambridge Univ. Press, Cambridge, 1996,
pp. 439–449.
[PSSW07] Y. Peres, O. Schramm, S. Sheffield, and D. Wilson, Random-turn hex and other selection
games, Amer. Math. Monthly 114 (2007), no. 5, 373–387.
[Sny08]
N. Snyder, Bidding hex, http://sbseminar.wordpress.com/2008/11/28/bidding-hex/, November
2008.
[Wil]
D. Wilson, Hexamania, http://research.microsoft.com/∼dbwilson/hex/.